Robust Topology Optimization of Periodic Multi-Material Functionally Graded Structures under Loading Uncertainties

This paper presents a robust topology optimization design approach for multi-material functional graded structures under periodic constraint with load uncertainties.To characterize the random-field uncertainties with a reduced set of random variables,the Karhunen-Lo`eve(K-L)expansion is adopted.The sparse grid numerical integration method is employed to transform the robust topology optimization into a weighted summation of series of deterministic topology optimization.Under dividing the design domain,the volume fraction of each preset gradient layer is extracted.Based on the ordered solid isotropic microstructure with penalization(Ordered-SIMP),a functionally graded multi-material interpolation model is formulated by individually optimizing each preset gradient layer.The periodic constraint setting of the gradient layer is achieved by redistributing the average element compliance in sub-regions.Then,the method of moving asymptotes(MMA)is introduced to iteratively update the design variables.Several numerical examples are presented to verify the validity and applicability of the proposed method.The results demonstrate that the periodic functionally graded multi-material topology can be obtained under different numbers of sub-regions,and robust design structures are more stable than that indicated by the deterministic results.

A Comparative Study of Stochastic Collocation Methods for Flow in Spatially Correlated Random Fields

Stochastic collocation methods as a promising approach for solving stochastic partial differential equations have been developed rapidly in recent years.Similar to Monte Carlo methods,the stochastic collocation methods are non-intrusive in that they can be implemented via repetitive execution of an existing deterministic solver without modifying it.The choice of collocation points leads to a variety of stochastic collocation methods including tensor product method,Smolyak method,Stroud 2 or 3 cubature method,and adaptive Stroud method.Another type of collocation method,the probabilistic collocation method(PCM),has also been proposed and applied to flow in porous media.In this paper,we discuss these methods in terms of their accuracy,efficiency,and applicable range for flow in spatially correlated random fields.These methods are compared in details under different conditions of spatial variability and correlation length.This study reveals that the Smolyak method and the PCM outperform other stochastic collocation methods in terms of accuracy and efficiency.The random dimensionality in approximating input random fields plays a crucial role in the performance of the stochastic collocation methods.Our numerical experiments indicate that the required random dimensionality increases slightly with the decrease of correlation scale and moderately from one to multiple physical dimensions.

A Comparison of Deterministic, Reliability-Based Topology Optimization under Uncertainties

Reliability and optimization are two key elements for structural design. The reliability~ based topology optimization （RBTO） is a powerful and promising methodology for finding the optimum topologies with the uncertainties being explicitly considered, typically manifested by the use of reliability constraints. Generally, a direct integration of reliability concept and topol- ogy optimization may lead to computational difficulties. In view of this fact, three methodologies have been presented in this study, including the double-loop approach （the performance measure approach, PMA） and the decoupled approaches （the so-called Hybrid method and the sequential optimization and reliability assessment, SORA）. For reliability analysis, the stochastic response surface method （SRSM） was applied, combining with the design of experiments generated by the sparse grid method, which has been proven as an effective and special discretization technique. The methodologies were investigated with three numerical examples considering the uncertainties including material properties and external loads. The optimal topologies obtained using the de- terministic, RBTOs were compared with one another; and useful conclusions regarding validity, accuracy and efficiency were drawn.